Compton scattering of twisted light: angular distribution and polarization of scattered photons
CCompton scattering of twisted light: angular distribution and polarization ofscattered photons
S. Stock,
1, 2, ∗ A. Surzhykov, S. Fritzsche,
1, 2 and D. Seipt † Helmholtz-Institut Jena, Fr¨obelstieg 3, 07743 Jena, Germany Friedrich-Schiller-Universit¨at Jena, Theoretisch-Physikalisches Institut, 07743 Jena, Germany
Compton scattering of twisted photons is investigated within a non-relativistic framework usingfirst-order perturbation theory. We formulate the problem in the density matrix theory, whichenables one to gain new insights into scattering processes of twisted particles by exploiting thesymmetries of the system. In particular, we analyze how the angular distribution and polarizationof the scattered photons are affected by the parameters of the initial beam such as the openingangle and the projection of orbital angular momentum. We present analytical and numerical resultsfor the angular distribution and the polarization of Compton scattered photons for initially twistedlight and compare them with the standard case of plane-wave light.
PACS numbers: 42.50.Tx, 03.65.NkKeywords: Compton scattering, twisted photons, optical vortices, density matrix formalism
I. INTRODUCTION
The inelastic scattering of photons on (quasi-)freecharged particles, known also as the Compton scatter-ing of light, is one of the best studied processes in quan-tum mechanics. This process demonstrates that light ismore than a classical wave phenomenon and that quan-tum theory is required in order to explain the frequencyshifts as well as the angular and polarization distribu-tion of the scattered light [1, 2]. For example, the classi-cal electro-magnetic theory cannot properly describe thefrequency shifts at low intensity of the incident photons[3], although these shifts are derived quite easily fromthe conservation of the (total) energy and momentum ofthe quantum particles involved in the scattering. In theframework of quantum theory, a shift in the wavelengthof the photons occurs since, for an electron at rest, forexample, a part of the incident photon energy is trans-ferred to the recoil of the electron. Therefore, an elasticscattering of the light can only be assumed if the photonenergy is negligible, compared to the rest energy of theelectrons. This low-energy limit to the Compton scatter-ing of photons can be described by the non-relativisticSchr¨odinger theory [4].Indeed, derivations of the angular distribution and po-larization of the Compton scattered light can be foundin many texts but are made usually for plane-wave pho-tons and electrons [4, 5]. Such a plane-wave approxima-tion to the Compton scattering applies if the lateral sizesof the incident electron and photon beams are (much)larger than their wavelength. In contrast, less attentionhas been paid to the Compton scattering of “twisted”beams in which each photon (or electron) carries an non-zero projection of the orbital angular momentum (OAM)along the propagation direction, in addition to the spin ∗ [email protected] † [email protected] angular momentum that is related to the polarization ofthe light [6–10]. Here, we therefore investigate how anOAM of the incident radiation affects the angular dis-tribution and polarization of the (Compton) scatteredlight. In particular, we apply the density matrix for-malism [11] to explore the angular distribution and po-larization of the scattered light and compare this to theresults of a plane-wave scattering. Calculations of theangle-differential Compton cross sections have been per-formed for the scattering of Bessel beams with differentopening angles and total angular momenta m . While thecross section does not depend on m , it is highly sensi-tive with regard to the opening angle θ k of the Besselbeams. Moreover, the formulation of the problem withinthe density matrix theory clearly highlights why the re-sults do not depend on m . This can be explained asrestriction to those elements of the twisted state pho-tonic density matrix that are diagonal in the momentumquantum numbers due to the spatial symmetries of thesystem.This work is structured as follows. In Sec. II, we brieflyreview the non-relativistic description of Compton scat-tering in the framework of the density matrix theory.This includes a short account on the electron-photon in-teraction in the non-relativistic framework as well as thequantization of the photon field. We derive the densitymatrix of the Compton scattered photons in terms of theusual plane-wave matrix elements which are weighted bythe initial photonic density matrix in a plane-wave basis.The standard case of plane-wave Compton scattering ispresented for later reference in Sec. III. The descriptionof twisted photon states is presented in Sec. IV. Detailedcalculations have been performed for the angular distri-bution and polarization of Compton scattered twistedphotons, and will be explained in comparison with theplane-wave case. A short summary is finally given inSec. V. In the Appendix, in addition, we collect all is-sues related to the normalization of the plane-wave andtwisted-wave one-particle states and their density matri-ces. a r X i v : . [ phy s i c s . a t o m - ph ] M a y II. DENSITY MATRIX THEORY FORNON-RELATIVISTIC COMPTON SCATTERING
We consider the scattering of plane-wave or twistedphotons on either a beam of free electrons as emittedfrom an electron gun [12], or on a target material witha low work function, much lower than the frequency oflight, such that the electrons can be considered as quasi-free. Twisted light has been produced in a wide range offrequencies [9, 13–17]. If the frequency of the photon ω in the rest frame of the electron is much smaller than theelectron rest energy, i.e. if the recoil parameter r = ̵ hωm e c ≪ , (1)we can work in the low-energy limit of non-relativisticCompton scattering. We then conveniently work in therest frame of the incident electron, where the theoret-ical description of the scattering process is much eas-ier. For an electron beam target the results observed inthe laboratory frame where the electrons are moving canbe obtained by just performing a proper Lorentz trans-formation. For electron beams with low kinetic energy, E kin ≲ ̵ h = α = e . A. Density Matrix Formalism
To describe the angular distribution and polarizationof the scattered photons it is most convenient to use thedensity-matrix theory [11, 20]. The density matrix for-malism has been applied just recently to describe the in-teraction of twisted light with many-electron atoms andions [21]. In this formalism, the system after the collisionis described by the final state density operator ˆ ρ f , whichis related to the density operator of the system in theinitial state before the scattering, ˆ ρ i , by the scatteringoperator, ˆ ρ f = ˆ S ˆ ρ i ˆ S † , (2)and where ˆ S characterizes the interaction of the particlesduring the collision.Before the scattering, the electrons and photons areinitially independent and uncorrelated. The density op-erator of the initial state can thus be written as the directproduct of the electronic (ˆ ρ el i ) and photonic (ˆ ρ ph i ) opera-tors [21] ˆ ρ i = ˆ ρ el i ⊗ ˆ ρ ph i . (3)We describe the initial electron as a plane-wave in apure quantum state ∣ p i ⟩ with the density operator ˆ ρ el i = ∣ p i ⟩⟨ p i ∣ . The initial photon is described by the initialstate photonic density operator ˆ ρ γ ≡ ˆ ρ ph i ( γ ) , where γ refers to a set of quantum numbers to describe that state.Below, we will specify the quantum numbers γ that areneeded to represent either a plane-wave or a twisted-wavephoton.Let us now write the final state density operator ˆ ρ f in a matrix representation in a plane-wave basis, where ∣ f ⟩ = ∣ p f , k f Λ f ⟩ abbreviates the final plane-wave elec-tron and photon states, where p f denotes the final elec-tron momentum while k f and Λ f stand for the finalphoton momentum and helicity, respectively. Let usalso introduce complete sets of initial plane-wave states ∣ i ⟩ = ∣ p , k λ ⟩ , with ⨋ i ∣ i ⟩⟨ i ∣ = ⟨ f ∣ ˆ ρ f ∣ f ′ ⟩ =⨋ λ,λ ′ ̃ d k ̃ d k ′ ⟨ k λ ∣ ˆ ρ γ ∣ k ′ λ ′ ⟩⟨ p i , k ′ Λ ′ ∣ ˆ S † ∣ f ′ ⟩⟨ f ∣ ˆ S ∣ p i , k Λ ⟩ , (4)where we employed the fact that the initial electron is ina plane wave state as discussed above, and where ̃ d k isan abbreviation for the properly normalized integrationmeasure (for details we refer to the Appendix). Thisequation states that in order to calculate the final statedensity matrix for an arbitrary initial photon state ˆ ρ γ ,either plane-wave or a twisted-wave or any other photonstate, we just need to know the ordinary plane-wave S matrix elements to describe the physics. Equation (4)describes how these plane-wave S matrix elements haveto be weighted by the elements of the initial photonicdensity matrix in the plane-wave basis ⟨ k λ ∣ ˆ ρ γ ∣ k ′ λ ′ ⟩ .The elements of the S matrix themselves can be rep-resented in a form ⟨ f ∣ ˆ S ∣ i ⟩ = − πiδ ( E i − E f )⟨ f ∣ ˆ T ∣ i ⟩ , (5)where the delta function ensures the conservation of en-ergy, i.e. the total energy of the initial state particles E i equals the total energy of the final state particles E f .The matrix elements of the transition operator ˆ T can becalculated using perturbation theory [4].The normalization of the density matrix is purely con-ventional and for us it is most convenient to normalizeit to the total cross section. This is achieved by dividingout the flux of incident particles [22]: σ ∶= tr ˆ ρ f T V tr ( ˆ J ˆ ρ i ) , (6)where T and V are the interaction time and volume,respectively. In order to calculate the cross section fortwisted particles, which are spatially localized, we need aconvenient definition of flux density operator ˆ J in termsof the densities of colliding particles times their relativevelocity [23]. Since twisted beams are not spatially ho-mogeneous perpendicular to their propagation direction,a proper definition of the cross section according to thebook of Taylor [22] includes an average over the lateralstructure of the beam of incident particles. We con-clude that we should define the cross section by meansof the spatially averaged density of the initial particles ⟨ n ⟩ ∶= V − ∫ d x n ( x ) that is proportional to the traceof the initial state density operator ⟨ n ⟩ = tr ˆ ρ / V . Thus,the operator of the averaged density is just the unityoperator divided by the quantization volume V (see theAppendix), and the flux density operator ˆ J in (6) can berepresented as ˆ J = c ˆ1 el V ⊗ ˆ1 ph V , i.e. by the relative velocity,the speed of light c , times the operators of the averagedparticle densities of electrons and photons in the initialstate.Using the basis expansion of the density operator,Eq. (4), and by employing the energy conservation in(5) we find that the cross section (6) contains a factor δ ( E f − E i ) δ ( E f − E i ′ ) = δ ( E f − E i ) δ ( E i − E i ′ ) . For anon-zero contribution to the scattering cross section, thetotal energies of the initial state bases used for the ex-pansion of the final state density matrix, Eq. (4), needto be equal. Finally, we can express the scattering crosssection as [24] σ = πVc tr ˆ ρ γ ⨋ f ⨋ λ,λ ′ ̃ d k ̃ d k ′ δ ( E i − E f )× ⟨ k λ ∣ ˆ ρ γ ∣ k ′ λ ′ ⟩⟨ p i , k ′ λ ′ ∣ ˆ T † ∣ f ⟩⟨ f ∣ ˆ T ∣ p i , k λ ⟩ . (7)Here we already employed the normalization of the initialelectron plane-wave states tr ˆ ρ el i = ⟨ p i ∣ p i ⟩ =
1. This con-cludes our general discussion of the density matrix for-malism and the definition of the cross section. The mainresult of this section is the representation of the final-state density matrix (4) of Compton scattered light foran incident plane-wave or twisted-wave photon beam—characterized by the photonic density operator ˆ ρ γ —interms of the plane-wave matrix elements for the interac-tion of electrons and photons. B. Interaction Between Photons and Electrons
Let us now turn our attention on the description of theinteraction between the electrons and the photons. Theform of the interaction Hamiltonian,ˆ H int = e ˆ A · ˆ p m e c + e ˆ A m e c , (8) follows from the gauge invariant minimal coupling of theelectromagnetic field to the free electron Hamiltonian[4]. The operator ˆ A of the electromagnetic vector po-tential describes the emission or the absorption of onephoton [4]. Compton scattering is a two-photon pro-cess: The incident photon is absorbed by the electronwhile the scattered photon is emitted into some other di-rection. The one-photon interaction operator ˆ A · ˆ p doesnot contribute to the non-relativistic Compton scatteringamplitude, because the matrix elements of the electronmomentum operator ˆ p vanish in the rest frame of theelectron [4]. Thus, the Compton scattering amplitudecan be calculated in first-order perturbation theory bymeans of the two-photon contribution ˆ A to the Hamil-tonian; the transition matrix elements are just given by ⟨ f ∣ ˆ T ∣ i ⟩ = ⟨ f ∣ ˆ H int ∣ i ⟩ .The photon field operator ˆ A that enters the interac-tion Hamiltonian, Eq. (8), can be represented by its modeexpansion into a circularly polarized plane-wave basis u k Λ ( x ) = e i k · x ε k Λ , where the polarization vector ε k Λ is perpendicular to the wave-vector, k · ε k Λ =
0. Thereare two independent solutions for each k , denoted by thephoton helicity Λ = ±
1. In terms of these plane-wavemodes the photon field operator is given byˆ A ( x ) = ⨋ Λ ̃ d k N k [ ˆ c k Λ u k Λ ( x ) + ˆ c † k Λ u ∗ k Λ ( x )] , (9)where we employ the proper integration measure ̃ d k to“count” the basis functions (see the Appendix for de-tails). The creation operator ˆ c † k Λ creates a normalizedone-photon plane-wave state from the vacuum ∣ k Λ ⟩ = ˆ c † k Λ ∣ ⟩ that is characterized by its linear momentum(wave-vector) k and helicity Λ. The one-photon statesare normalized as ⟨ k Λ ∣ k Λ ⟩ =
1. Moreover, the normal-ization factor N k = √ πc / kV is determined such thatthe energy eigenvalues of the free field Hamiltonian arejust ω k = c ∣ k ∣ for one-photon states ∣ k Λ ⟩ [4]. It is suf-ficient to know the above representation of the photonfield operator in a plane-wave basis because we just needto calculate the plane-wave matrix elements by means ofEqs. (4) and (7). C. The Reduced Density Matrix of ComptonScattered Photons
We are going to investigate the angular distributionand polarization of the scattered photons, and we arenot interested in the electron distribution after the scat-tering has occurred. We therefore have to calculate areduced density matrix ρ Λ ′ Λ ( k f / k f ) , which depends onthe direction of the scattered photon k f / k f and its po-larization state Λ, by tracing out the unobserved finalelectron states. We obtain the reduced density matrix ofthe Compton scattered photons ρ Λ ′ Λ ( k f / k f ) = V ( π ) c tr ˆ ρ γ ⨋ λ,λ ′ ̃ d k ̃ d k ′ ∫ ̃ d p f ∫ d k f k f δ ( E f − E i ) ⟨ k λ ∣ ˆ ρ γ ∣ k ′ λ ′ ⟩ M ∗ k ′ λ ′ ( Λ ′ )M k λ ( Λ ) , (10)in terms of the plane-wave matrix elements M k λ ( Λ ) =⟨ p f ; k f Λ ∣ ˆ H int ∣ p i ; k λ ⟩ . The calculation of the plane-wavematrix elements for non-relativistic Compton scatteringcan be found in textbooks, e.g. in [4], and we only citehere the final result: M k λ ( Λ ) = e m e c πc L ε ∗ k f Λ · ε k λ √ ω i ω f ̃ δ ( p i + k − k f − p f ) . (11)Here, ˜ δ denotes a normalized delta function with theproperty ˜ δ ( ) =
1. For details we refer to the Appendix.Because the reduced density matrix (10) contains theproduct of two plane-wave matrix elements we also gettwo delta functions that ensure the conservation of mo-mentum. Their product can be reformulated as, ̃ δ ( p i + k − p f − k f )̃ δ ( p i + k ′ − p f − k f )= ̃ δ ( k ′ − k )̃ δ ( p f + k f − p i − k ) , (12)where we obtain a factor ̃ δ ( k ′ − k ) , which consumes oneof the integrations over the plane-wave bases, ̃ d k ′ , inEq. (10). For this reason, only those elements of the ini-tial state photonic density matrix with k = k ′ remain inEq. (10). Thus, just the momentum-diagonal elements of ˆ ρ γ contribute to the reduced density matrix of thescattered photons and the coherences in the off-diagonalelements of the initial photonic density matrix are lost.The reason for this behaviour is of course the momen-tum conservation in the plane-wave matrix elements thatis related to the spatial homogeneity of the system viaNoether’s theorem [25]. All particles are described asplane-waves, except for the initial photons that are pre-pared in the hitherto unspecified quantum state ˆ ρ γ . It is the spatial homogeneity of the residual system of incidentand scattered particles which excludes the interference ofdifferent momentum components of the initial photonicstate ˆ ρ γ from the reduced density matrix (10) of the scat-tered photons.The above momentum conservation, together with theconservation of the total energy δ ( E f − E i ) = δ ( ω f + p f / m e − ω − p i / m e ) , determines the frequency of thescattered photons. Recalling that we work in the restframe of the incident electron, p i =
0, we find for thefrequency of the scattered photons ω f = ω [ − ̵ hωm e c ( − cos θ )+ O ( ̵ hωm e c ) ] , (13)where we temporarily reinstated ̵ h . The expression for ω f in (13) accounts for the well-known frequency red-shift of Compton scattering [1], which depends on theangle θ between the initial and the scattered photon mo-mentum vectors. When starting from a fully relativisticQED calculation, we would get the above non-relativisticfrequency shift in (13) the leading order of an expan-sion in the small recoil parameter r , Eq. (1). Because ofEq. (13), the length of the scattered photon’s wave-vector ∣ k f ∣ = ω f ( θ )/ c is completely determined by its direction.In the electric dipole approximation, the momentum ofthe photon and its recoiling effect on the electron is ne-glected, k = k f =
0. This is a good approximation when-ever the recoil parameter, Eq. (1), is negligibly small.The non-relativistic Compton scattering becomes elasticwithin the dipole approximation: ω f = ω . Moreover, thedipole approximation coincides with the formal classicallimit ̵ h →
0. Within the dipole approximation we obtainas final formula for the elements of the reduced densitymatrix the following expression: ρ Λ ′ Λ ( k f / k f ) = e m c ⨋ λ,λ ′ ̃ d k ⟨ k λ ∣ ˆ ρ γ ∣ k λ ′ ⟩ tr ˆ ρ γ ( ε ∗ k f Λ ′ · ε k λ ′ ) ∗ ( ε ∗ k f Λ · ε k λ ) . (14) D. Stokes parameters and differential cross section
In the previous subsection we calculated a suitable ex-pression for the reduced density matrix of Compton scat-tered photons, Eq. (14). We now relate the elements ofthe reduced density matrix to the angular distribution of the scattered photons, i.e. to their angular differentialcross section and the polarization properties. Accordingto [11, 20], the reduced density matrix can be representedby the three Stokes parameters P = ( P , P , P ) via ρ Λ ′ Λ ( k f ) = d σ dΩ 12 ( + P P − iP P + iP − P ) Λ ′ Λ . (15)From this representation it is easy to obtain the angulardifferential cross section of Compton scattered photonsas d σ dΩ = ∑ Λ =± ρ ΛΛ ( k f ) = ρ + + + ρ − − . (16)The Stokes parameters are given by P = ρ + − + ρ − + ρ + + + ρ − − , (17) P = iρ + − − iρ − + ρ + + + ρ − − , (18) P = ρ + + − ρ − − ρ + + + ρ − − . (19)As usual, the Stokes parameters P and P represent theintensity of light that is linearly polarized under differentangles with respect to the scattering plane. The scatter-ing plane is defined by the direction of the incident beamof light and by the momentum vector of the scatteredphoton. The Stokes parameter P measures the amountof light with circular polarization [20, 26]. Moreover, thedegree of polarization Π is defined as the length of thevector P Π = √ P + P + P . (20)This concludes our discussion of the density matrix for-malism. We are now ready to study the angular distri-bution and the polarization properties of Compton scat-tered light for both plane-wave and twisted photons. III. COMPTON SCATTERING OFPLANE-WAVE PHOTONS
Let us now apply the formalism to the standard caseof the Compton scattering of plane-wave photons, asa starting point for later comparison with the case oftwisted light. Moreover, this will convince us that wenormalized the reduced density matrix correctly to ob-tain the differential cross section by comparing with thewell known results from the literature.We now specify the initial photon state as a planewave with wave-vector k i and in a well defined helic-ity state Λ i , with the photonic initial density operatorˆ ρ γ = ˆ ρ k i Λ i = ∣ k i Λ i ⟩⟨ k i Λ i ∣ . It has the following represen-tation in a plane-wave basis: ⟨ k λ ∣ ˆ ρ k i Λ i ∣ k ′ λ ′ ⟩ = ̃ δ ( k − k ′ ) δ λλ ′ ̃ δ ( k − k i ) δ λ Λ i . (21)Using its diagonal elements ⟨ k λ ∣ ˆ ρ k i Λ i ∣ k λ ′ ⟩ = δ λλ ′ δ λ Λ i ̃ δ ( k − k i ) into Eq. (14) readily yields thereduced density matrix of non-relativistic Comptonscattering of circularly polarized plane-wave photons inthe dipole approximation ρ Λ ′ Λ ( k f ) = r ( ε ∗ k f Λ ′ · ε k i Λ i ) ∗ ( ε ∗ k f Λ · ε k i Λ i ) , (22) where r e = e /( m e c ) ≃ . z -axis, k i / k i = ( , , ) T , while the scattered photon propagatesinto the direction k f / k f = ( sin θ cos ϕ, sin θ sin ϕ, cos θ ) T ,where the scattering angle θ and azimuthal angle ϕ arethe usual polar and azimuthal angles in spherical coordi-nates. For the polarization vectors of the scattered pho-tons we give the explicit representation ε k f Λ = ε Λ ( θ, ϕ ) = √ ⎛⎜⎝ cos θ cos ϕ − i Λ sin ϕ cos θ sin ϕ + i Λ cos ϕ − sin θ ⎞⎟⎠ , (23)which makes evident that the polarization vector is or-thonormalized ε ∗ k f Λ · ε k f Λ ′ = δ ΛΛ ′ and perpendicular tothe momentum direction k f · ε k f Λ =
0. Because the inci-dent photon propagates along the z -axis, its polarizationvector is just ε k i Λ i = ε Λ i ( , ) = ( /√ , i Λ i /√ , ) T .The differential Compton cross section as a functionof the scattering angle is just given by using the repre-sentations (23) of the photons polarization vectors intoEq. (22), and by summing over the final state polariza-tion according to Eq. (16). This yieldsd σ pw dΩ = r ( + cos θ ) , (24)where θ denotes the scattering angle, i.e. the angle be-tween the wave-vectors of the incident and the scat-tered light. The result (24) for the angular differentialcross section of Compton scattered light in the dipoleapproximation is well known and can also be obtainedby means of classical electrodynamics [3]. An integrationof Eq. (24) over all directions of the scattered photonsjust yields the well known total Thomson cross section σ = πr ≃
665 mb. This comparison with well-knownresults from the literature shows that we normalized thefinal state density matrix correctly to the cross section.Similarly to the cross section, we can obtain explicitexpressions for the three Stokes parameters of the scat-tered photons as P = − sin θ + cos θ , (25) P = , (26) P = i cos θ + cos θ . (27)As we see from Eqs. (25) – (27), the scattered photonsare not necessarily circularly polarized, although the ini-tial photons were. The ratio of the amount of linearlyand circularly polarized photons varies with the scatter-ing angle θ . For instance, under θ = ○ we have P = − = IV. COMPTON SCATTERING OFTWISTED-WAVE PHOTONS
We now turn to the main aspect of our paper: Thecalculation of the angular distribution and the polariza-tion properties of Compton scattered twisted light. Forthat, we need first to construct the initial photonic den-sity matrix for twisted photons.
A. Description of Twisted Photon States and theTwisted Photon Density Matrix
The state of a photon in a Bessel beam that propagatesalong the z -axis, briefly referred to as a Bessel-state ortwisted photon, is characterized by its longitudinal mo-mentum κ ∥ , i.e. the component of the linear momentumalong the beam’s propagation axis, the modulus of thetransverse momentum κ ⊥ = ∣ k ⊥ ∣ , the projection of thetotal angular momentum (TAM) onto the propagationaxis m , and the photon helicity Λ [27]. A photon in atwisted one-particle Bessel-state is, thus, characterizedby the quantum numbers ∣ γ ⟩ = ∣ κ ⊥ κ ∥ m Λ ⟩ . It can be rep-resented by a coherent superposition of plane-wave states[21, 27–29], ∣ κ ⊥ κ ∥ m Λ ⟩ = ∫ ̃ d k b κ ⊥ κ ∥ m ( k ) ∣ k Λ ⟩ , (28)where we use the proper integration measure ̃ d k forplane-wave states (see the Appendix). From Eq. (28) wesee that Λ refers to the helicity of the plane-wave compo-nents of the twisted photon. The amplitudes b κ ⊥ κ ∥ m ( k ) are given by b κ ⊥ κ ∥ m ( k ) = N tw δ ( k z − κ ∥ ) a κ ⊥ m ( k ⊥ ) (29)and are related to the usually employed transverse am-plitudes (see e.g. [21, 27, 28]) a κ ⊥ m ( k ⊥ ) = √ πκ ⊥ (− i ) m e imϕ k δ (∣ k ⊥ ∣ − κ ⊥ ) . (30)Because for a photon in a twisted Bessel state κ ∥ and κ ⊥ are well-defined, all the momentum vectors k of the su-perposition (28) are lying on a cone in momentum spacewith fixed opening angle θ k = arctan ( κ ⊥ / κ ∥ ) . The direc-tion of the momentum vector k on the cone is undefined,and can be parametrized as k = k ( ϕ k ) = ⎛⎜⎝ κ ⊥ cos ϕ k κ ⊥ sin ϕ k κ ∥ ⎞⎟⎠ = k ⎛⎜⎝ sin θ k cos ϕ k sin θ k sin ϕ k cos θ k ⎞⎟⎠ . (31) where ϕ k is the azimuthal angle that defines the orien-tation of one particular vector k ( ϕ k ) on the momen-tum cone. The length of these vectors, k = ∣ k ( ϕ k )∣ =√ κ ∥ + κ ⊥ , are related to the photon frequency ω = ck asfor plane-waves.The normalization factor N tw = √ π / L z RV that ap-pears in the definition of the amplitudes b κ ⊥ κ ∥ m Λ is de-termined such that the twisted one-particle states areorthonormalized in the following way: ⟨ κ ⊥ κ ∥ m Λ ∣ κ ′⊥ κ ′∥ m ′ Λ ′ ⟩= π RL z δ mm ′ δ ΛΛ ′ δ ( κ ⊥ − κ ′⊥ ) δ ( κ ∥ − κ ′∥ ) (32)and ⟨ κ ⊥ κ ∥ m Λ ∣ κ ⊥ κ ∥ m Λ ⟩ =
1. This corresponds to thenormalization to one particle per cylindrical volume V = πR L z , where both the radius R and the length L z ofthe cylinder are going to infinity (see the Appendix).Let us now construct the density operator for twistedphotons in the pure quantum state (28), together with itsmatrix representation in a plane-wave basis. The latter isneeded to calculate the reduced density matrix, Eq. (14),of Compton scattered twisted light. The normalizationof the one-photon states implies that the twisted-statedensity operatorˆ ρ γ = ˆ ρ κ ⊥ κ ∥ m Λ = ∣ κ ⊥ κ ∥ m Λ ⟩⟨ κ ⊥ κ ∥ m Λ ∣ (33)has unity trace, tr ( ˆ ρ κ ⊥ κ ∥ m Λ ) =
1, i.e. it is normalized toan average particle density of “one particle per volume V ”. The matrix elements of the twisted density operator(33) in a plane-wave basis are just given by products ofthe amplitudes b κ ⊥ κ ∥ m Λ , Eq. (29), ⟨ k λ ∣ ˆ ρ κ ⊥ κ ∥ m Λ ∣ k ′ λ ′ ⟩ = δ λλ ′ δ λ Λ b κ ⊥ κ ∥ m ( k ) b ∗ κ ⊥ κ ∥ m ( k ′ )∝ δ λλ ′ δ λ Λ e im ( ϕ k − ϕ ′ k ) (34)and they are diagonal in the helicity quantum numbers.We stress that only the momentum-off-diagonal elementsof the density matrix (34) do depend on the projectionof total angular momentum m . It enters as the differenceof the vortex phase factors e imϕ k of the two plane-wavecomponents k ≠ k ′ . On the other hand, the momentum-diagonal elements of the above density matrix ⟨ k λ ∣ ˆ ρ κ ⊥ κ ∥ m Λ ∣ k λ ′ ⟩= δ λλ ′ δ λ Λ ( π ) V κ ⊥ δ ( k z − κ ∥ ) δ (∣ k ⊥ ∣ − κ ⊥ ) (35)which enter the calculation of the reduced density matrixof Compton scattered photons, Eq. (14), are completelyindependent of m . Therefore, also the angular distribu-tion and polarization of the Compton scattered photonsdo not dependent of m .A few remarks might be in order why the reduceddensity matrix of the scattered photons is independentfrom m . In this paper we are interested in the angu-lar distribution and polarization properties of the Comp-ton scattered photons. We, therefore, project the finalstate density operator onto a basis of plane-wave statesthat can be observed by a usual detector, which mea-sures the linear momentum of a photon [29]. Thus, allinvolved particles except for the initial twisted photonsare described as plane-wave states. From the discus-sion in the previous section we know that the momen-tum conservation described by the delta function in theplane-wave matrix elements enforces the restriction tothe momentum-diagonal elements of the initial photonicdensity matrix. Thus, we loose the dependence on m because only the momentum-diagonal elements of thephotonic density matrix (34) that describes the initialtwisted state contribute due to symmetries of the sys-tem.It is known from previous studies that the scatteringof twisted particles on spatially homogeneous systems,such as plane-waves [29, 30], impact-parameter averagedatomic targets [21, 31], impact-parameter averaged po-tential scattering [32], leads to angular distributions ofthe scattered particles or fluorescence light that are in-dependent of m . On the other hand, the coherences ofthe initial state density matrix will play a role for scenar-ios with a spatial inhomogeneity other than the twistedbeam. For instance the angular distributions do dependon m for the collision of a twisted particle with an in-homogeneous target, like: a second beam of twisted par-ticles [29, 33], a localized microscopic target such as asingle atom [21, 27, 34–36], or a quantum dot [37].Another possibility to recover the coherences in theoff-diagonal elements of the twisted density matrix is tolook for the angular momentum of the scattered parti-cles. In fact, it has been shown in [28, 38] that Comptonbackscattered photons do indeed carry orbital angularmomentum. In order to access the angular momentum ofthe scattered photons, one needs to determine the final-state density operator ˆ ρ f in the the basis of the twistedstates. This requires a suitable detection operator thatdirectly measures the orbital angular momentum of thescattered photons [20, 28, 29]. B. Angular distribution and Stokes parameters forthe scattering of a twisted photon with well-definedTAM
If we substitute the initial state density matrix (35)of the twisted photon into Eq. (14), and by performingthe integration over the plane-wave basis in cylindricalcoordinates ∫ ̃ d k = V ( π ) ∫ d ϕ k d k z d k ⊥ k ⊥ , we obtain thereduced density matrix for the Compton scattering of atwisted photon (in the dipole approximation ω f = ω i ) . Itincludes an integration over all plane-wave components as described by ϕ k ρ Λ ′ Λ = r ∫ d ϕ k π ( ε ∗ k f Λ ′ · ε k ( ϕ k ) Λ i ) ∗ ( ε ∗ k f Λ · ε k ( ϕ k ) Λ i ) , (36)with k ( ϕ k ) from Eq. (31). This reduced density matrixcan be directly compared with the corresponding resultfor plane-wave light, Eq. (22). . . . . .
91 0 20 40 60 80 100 120 140 160 180 d σ t w / d Ω (cid:2) r e (cid:3) Scattering angle θ [ ◦ ] θ k = 0 ◦ θ k = 30 ◦ θ k = 60 ◦ θ k = 90 ◦ Figure 1. (Color online) Differential cross section d σ tw / dΩfor Compton scattering of twisted light as a function of thescattering angle θ . Results are shown for various values of themomentum cone opening angle θ k of the twisted photons. Thecase θ k = θ k → ○ of large cone opening angles can not bereached experimentally. If we make use of the explicit representation of the po-larization vectors (23), the integration over the azimuthalangle ϕ k in Eq. (36) can be carried out and yields the dif-ferential cross sectiond σ tw dΩ = r [( + cos θ k )( + cos θ ) + θ k sin θ ] , (37)where θ denotes the scattering angle measured from the z -axis, and θ k = arctan κ ⊥ / κ ∥ denotes the opening angleof the initial twisted photon beam. As anticipated above,the differential cross section is independent of the valueof the projection of total angular momentum m , but itdoes depend on the momentum cone opening angle θ k .Note that in the limit θ k → θ for var-ious cone angles θ k . The red solid curve ( θ k =
0) cor-responds to the case of initial plane-wave photons andshows the well known symmetric angular distributionwhich is minimal at the scattering angle θ = ○ , whereit is just 1 / θ k > θ = ○ becomes lesspronounced. For sufficiently large θ k > θ ⋆ k the distribu-tion turns around with the maximum of the angular dis-tribution occurring at θ = ○ (e.g. for the blue dottedcurve in Fig. 1). This crossover occurs at the “magicangle” of θ ⋆ k = arccos ( /√ ) ≈ . ○ , where the angulardistribution is flat.In addition to the angular distribution discussed above,the momentum cone opening angle θ k of the twisted pho-ton beam also influences the polarization properties ofthe Compton scattered light. For incident photons inthe Bessel state, the Stokes parameters are given by P = ( − θ k ) sin θ ( + cos θ k )( + cos θ ) + θ k sin θ , (38) P = , (39) P = i cos θ k cos θ ( + cos θ k )( + cos θ ) + θ k sin θ . (40)Similar to the differential cross section, the Stokes pa-rameters do neither depend on the azimuthal angle ofthe scattered photon, nor on the total angular momen-tum m of the incident twisted beam, as the whole sce-nario is cylindrically symmetric. Again the results forthe plane-wave case are reproduced for θ k = θ for differentopening angles θ k . As discussed above for plane-wavephotons, the values of P and P , which quantify the lin-ear and circular polarization of the scattered radiation,respectively, depend on the scattering angle. Both valuesare sensitive to the cone opening angle θ k of the twistedlight. For instance, as depicted in Fig. 2 (a), the value of P at 90 ○ scattering decreases in magnitude for increas-ing values of θ k , starting from P = − θ k = θ ⋆ k = arccos ( /√ ) the scattered photons are not linearlypolarized, since P = θ k > θ ⋆ k the value of P is positive which indicates achange of the plane of linear polarization of the scatteredphotons which are now (partially) polarized in the scat-tering plane. Since the sign of P depends on the helicityΛ i = ± P / Λ i instead. The values of P grad-ually decrease for increasing θ k , and approach zero for θ k → ○ . For twisted light with θ k >
0, the degree ofpolarization is smaller than 1, so the scattered photonsare not fully polarized anymore. For not too large coneopening angles θ k the scattered radiation is depolarizedthe strongest at θ = ○ . For sufficiently large θ k angulardependence of the degree of polarization shows the oppo-site behavior. In particular for θ k → ○ the scattered ra-diation becomes completely depolarized for forward andbackward scattering, while the degree of polarization Πis nonzero at finite scattering angles, see Fig. 2 (c). C. Angular distribution for the scattering of asuperposition of twisted photons
As discussed above, the angular distribution of Comp-ton scattered photons and their polarization does not de-pend on the value of the total angular momentum m ifthe initial light is prepared in a Bessel state with well-defined m ; it just depends on the opening angle θ k of thebeam.We now examine the Compton scattering of a coher-ent superposition of two states with equal longitudinaland transverse momenta, κ ⊥ and κ ∥ , equal helicity Λ,but with two different values of total angular momentum m > m . Such a superposition is described by the statevector ∣ γ ⟩ = c ∣ κ ⊥ κ ∥ m Λ ⟩ + c ∣ κ ⊥ κ ∥ m Λ ⟩ , (41)where the coefficients fulfil ∣ c ∣ + ∣ c ∣ = ∣ γ ⟩ is normalized ⟨ γ ∣ γ ⟩ =
1. The state vector (41) lies onthe Bloch sphere with the two basis states as the polesbecause ∣ γ ⟩ is a pure quantum state [9]. The experimentalgeneration of such superpositions of twisted states hasbeen reported, e.g., in [39].Such superpositions of twisted beams have been con-sidered previously in theoretical studies of scattering andatomic absorption processes, e.g. in Refs. [21, 29, 30]. Inthese previous studies it was seen already that the in-terference between these two TAM eigenstates leads toangular distributions of the scattered particles that de-pend on the difference ∆ m = m − m of the total an-gular momentum values of the two beams. This inter-ference between the two components of the photon state ∣ γ ⟩ , Eq. (41), can be seen in the off-diagonal elements ofthe density matrix of the initial photon state ∣ γ ⟩ in theplane-wave basis ⟨ k λ ∣( ˆ ρ γ ) m ′ m ∣ k λ ⟩ = δ λ Λ ( π ) κ ⊥ V δ ( k ⊥ − κ ⊥ ) δ ( k ∥ − κ ∥ )× ( ∣ c ∣ c c ∗ e − i ∆ m ( ϕ k − π / ) c ∗ c e i ∆ m ( ϕ k − π / ) ∣ c ∣ ) (42)The interferences are maximized by choosing coefficients c n with equal modulus and a relative phase δ as, e.g., c = /√ c = e iδ /√
2. These equal-weighted super-positions are all lying on the equator of the Bloch sphere,and where the relative phase δ = . . . π denotes the lon-gitude [9].Let us note here that still only the momentum-diagonalterms of the photonic density matrix ˆ ρ γ do contribute tothe scattering cross section. The superpostion of morethan one twisted state modifies the distribution of plane-wave components on the momentum cone so that theyare no longer uniformly distributed on the cone as forthe case of a single Bessel state. Instead, the azimuthaldistribution of plane wave modes is modulated by the dif-ference of the total angular momentum of the two beams∆ m . This is described by the off-diagonal elements in the (a) Stokes Parameter P (b) Stokes Parameter P (c) Degree of polarization Π − − . − . − . − . . . P Scattering angle θ [ ◦ ] − − . .
51 0 20 40 60 80 100 120 140 160 180 P / Λ i Scattering angle θ [ ◦ ] θ k = 0 ◦ θ k = 30 ◦ θ k = 60 ◦ θ k = 90 ◦ . . . . . D e g r ee o f p o l. Π Scattering angle θ [ ◦ ] Figure 2. (Color online) The polarization of Compton scattered photons is characterized by the Stokes parameters P (left), P (middle) and the degree of polarization Π (right) as a function of the scattering angle. Results are shown for beams oftwisted photons with different momentum cone opening angles θ k . As in Fig. 1, the case θ k = density matrix (42) in the space of the two basis statesof the superposition (41).For such a superposition we readily obtain the differ-ential cross section in the dipole approximationd σ dΩ = r ∑ Λ f ∫ d ϕ k π [ + cos ( ∆ m ( ϕ k − π ) + δ )]× ∣ ε ∗ Λ f ( θ, ϕ ) · ε Λ i ( θ k , ϕ k )∣ , (43)which can be split into two terms,d σ dΩ = d σ tw dΩ + d σ int dΩ . (44)The first term, with the “1” in the square brackets, is thecross section for the case of a single TAM eigenstate, al-ready discussed in the previous section, cf. Eq. (37). Thesecond term d σ int / dΩ describes the interference betweenthe two superimposed TAM eigenstates, and is related tothe off-diagonal elements of the density matrix in Eq. (42)by means of cos ( ∆ m ( ϕ k − π ) + δ ) , which describes theazimuthal modulation of the density of plane-wave stateson the momentum cone.By performing the integration over the azimuthal angle ϕ k the interference of these eigenstates contributes to thecross sectiond σ int dΩ = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ r sin ( ϕ + δ ) sin ( θ k ) sin ( θ ) , ∆ m = , − r cos ( ϕ + δ ) sin θ k sin θ , ∆ m = , , ∆ m ≥ . (45)if ∆ m = ω f ≠ ω i , Eq. (13), we find a non-vanishing interference contribution also for ∆ m = σ int dΩ = − ̵ hω i m e c r
16 sin ( ϕ + δ ) sin θ k sin θ , (46) which is proportional to the small recoil parameter ̵ hω i / m e c ≪
1, and is therefore much smaller than theinterference terms for ∆ m = , m = , θ k . Obviously, the interference termdepends on the azimuthal angle ϕ of the scattered pho-ton, and not on the scattering angle θ alone. The disks inFig. 3 represent a projection of the forward hemisphereof scattered photons, as drawn in the top left panel. Thesuperpositions of twisted photons break the axial symme-try of the initial state. The number of azimuthal modula-tions of plane-wave states on the momentum cone is just∆ m and the orientation of that pattern is determinedby the relative phase δ , i.e. by the latitude of the state ∣ γ ⟩ on the Bloch sphere. Changing the value of δ resultsin a rotation of the distributions of scattered photons inFig. 3 with respect to the azimuthal angle ϕ .Let us briefly discuss the angular distribution in thebackward-hemisphere, i.e. for scattering angles θ ≥ ○ .The distributions can be easily read off Eqs. (37) and(45). In particular, for the case ∆ m = σ ( ∆ m = ) / dΩ ( θ, ϕ ) = d σ ( ∆ m = ) / dΩ ( π − θ, − ϕ − δ ) .For ∆ m =
2, on the other hand, we obtain the symmetryrelation d σ ( ∆ m = ) / dΩ ( θ, ϕ ) = d σ ( ∆ m = ) / dΩ ( π − θ, ϕ ) . V. CONCLUSIONS
In summary, a theoretical study has been performedfor the Compton scattered of photons from a Bessel beamon electrons in the rest frame of the electrons. In thelong-wavelength limit of the incident radiation and basedon the non-relativistic Schr¨odinger’s equation, the den-sity matrix theory has been used to analyze the angle-differential cross section as function of the scattering an-gle and for Bessel beams with different cone opening an-gles and total angular momentum.Our formulation of the problem in the density matrix0 θ k = ○ θ k = ○ θ k = ○ θ k = ○ ∆ m = electrons k i k f (cid:2) r (cid:3) , , , , , , , , , , ∆ m = (cid:2) r (cid:3) , , , , , , , , , , Figure 3. (Color online) Angular differential cross section d σ / dΩ of Compton scattered light for initial photons in asuperposition of twisted waves with ∆ m = m = θ k , and for δ = θ for scattering angles 0 ≤ θ ≤ ○ , and the azimuthal axis represents the azimuthal angle ϕ ofthe scattered photon. For θ k = m = m = ϕ . formulation explains why the angular distribution andpolarization of the scattered photons do not depend onthe value of the projection of angular momentum: Be-cause of the symmetry of the system only the those ele-ments of the incident twisted photon density matrix con-tribute to the reduced density matrix of the scatteredphotons that are diagonal in the momentum quantumnumbers. The angular momentum value appears onlyvia the difference of the vortex phase factors of the plane-wave components of the twisted photon and, therefore,vanishes on the diagonal of the density matrix.We found completely analytical results for the angu-lar and polarization distribution of Compton scatteredphotons for the scattering of twisted photons. These dis-tributions are sensitive with regard to the momentumcone opening angle θ k . In particular we observed a de-polarization of the scattered radiation for large values ofthis cone opening angle. In addition, it was found thatthe angular distributions of the scattered photons for asuperpositions of twisted photon beams with different m differ from the case of a single TAM eigenstate if ∆ m = ∣ ∆ m ∣ > r in Eq. (1) is small, our results can be directly translated to scenarios of in-verse Compton scattering by applying a suitable Lorentztransformation. For inverse Compton scattering, low-frequency photons (e.g. optial laser photons) are scat-tered on ultra-relativistic electrons, and the backscat-tered photons’ frequency is Doppler up-shifted to the x-ray regime. Appendix: Normalization of quantum states ofplane-wave and twisted photons
In this Appendix we discuss all necessary details onthe normalization of the quantum states that enter thecalculation of the final-state density matrix in the non-relativistic Compton scattering of plane-wave or twistedlight. The usual normalization in a finite box does notwork in this case, because the twisted photons are cylin-drical symmetric modes. Moreover, the states of twistedphotons are represented as a continuous coherent super-position of plane waves, while in a finite-sized box the mo-mentum modes are discrete. Therefore, we need to quan-tize the modes in an infinite volume V → ∞ . In order tocorrectly normalize twisted-particle quantum states wewill explicitly keep all the factors of the formally infinitevolume. We normalize all one-particle quantum states ∣ ψ ⟩ , i.e. for the electrons and the plane-wave or twistedphotons, to unity: ⟨ ψ ∣ ψ ⟩ =
1. In the following we discusswhat this implies in detail for the electron and photonstates, their density operators, spatial wavefunctions, as1well as for the spatial probability density.
1. Plane-wave electron states
Throughout our paper, all initial and final states of theelectrons are described as plane waves, i.e. as momentumeigenstates ˆ p ∣ p ⟩ = p ∣ p ⟩ , where ˆ p is the electron momen-tum operator, and the states are characterized by thethree quantum numbers of the linear momentum eigen-value p = ( p x , p y , p z ) . We require that the one-particlestates are normalized as ⟨ p ∣ p ⟩ = , (A.1)and, hence, for the orthogonality relation ⟨ p ′ ∣ p ⟩ = ( πL ) δ ( p ′ − p ) , (A.2)and where the normalization (A.1) follows from (A.2)with the usual interpretation of lim p ′ → p δ ( p ′ − p ) =( L / π ) [40].We may define a regularized delta distribution ̃ δ ( p ′ − p ) ∶= ( πL ) δ ( p ′ − p ) , (A.3)which has the property lim p ′ → p ̃ δ ( p ′ − p ) =
1. The prop-erly normalized integration measure for these plane-wavestates is given by ∫ ̃ d p ∶= ∫ ( L π ) d p . (A.4)In particular, ∫ ̃ d p ̃ δ ( p − p ′ ) =
1. These definitions alsoprovide the correct way of counting the number of finalstates; their density is just ̃ d p . Moreover, the complete-ness relation for the plane-wave electron states is ∫ ̃ d p ∣ p ⟩⟨ p ∣ = ˆ1 . (A.5)Because the states ∣ p ⟩ are normalized, the trace of thedensity operator ˆ ρ p = ∣ p ⟩⟨ p ∣ is just unity tr ( ˆ ρ p ) = ψ p ( x ) = ⟨ x ∣ p ⟩ = e i p · x /√ V . The particle density is de-termined as the expectation value of the particle densityoperator ˆ n ( x ) = ∣ x ⟩⟨ x ∣ , as n p ( x ) = tr ˆ n ( x ) ˆ ρ p and justyields a constant local particle density of n p ( x ) = / V .Therefore one usually says that the plane-waves (A.1)are normalized to one particle in the infinite quantiza-tion volume V = L → ∞ .
2. Plane-wave photon states
We apply the same normalization to the plane-photonstates as we employed for the plane-wave electron states in the previous subsection. The only small differenceis that the photon states are characterized also by theirhelicity Λ in addition to the linear momentum eigenvalue k . We thus orthonormalize the plane-wave one particlephoton states ∣ k Λ ⟩ as ⟨ k Λ ∣ k ′ Λ ′ ⟩ = δ ΛΛ ′ ̃ δ ( k − k ′ ) . (A.6)Moreover, we quantize the photon field, expanding thephoton field operatorˆ A ( x ) = ⨋ Λ ̃ d k N k [ ˆ c k Λ u k Λ ( x ) + ˆ c † k Λ u ∗ k Λ ( x )] , (A.7)into a circularly polarized plane-wave basis u k Λ ( x ) = e i k · x ε k Λ , and with the same meaning of ̃ d k as for plane-wave electrons. The normalization factor N k = √ πc / kV is chosen in such a way that the free Hamiltonian of thephoton field is given by H γ = π ∫ d x ( ˆ E + ˆ B ) ! = ⨋ Λ ̃ d k ω k ( ˆ c † k Λ ˆ c k Λ + ) , (A.8)where ˆ E and B are the electric and magnetic field oper-ators, respectively.The one-particle states are generated by the photoncreation operators from the vacuum (zero-photon) state ∣ k Λ ⟩ = ˆ c † k Λ ∣ ⟩ . (A.9)The commutation relations of the photon creation andannihilation operators [ ˆ c k Λ , ˆ c † k ′ Λ ′ ] = δ ΛΛ ′ ̃ δ ( k − k ′ ) (A.10)completely fix the orthonormalization (A.6).The proper measure of the final states now includesa sum over the helicity states, thus, the completenessrelation is given by ⨋ Λ ̃ d k ∣ k Λ ⟩⟨ k Λ ∣ = ˆ1 . (A.11)That means we need to include the sum over the two he-licity states in the trace of the photonic density matrix inorder have tr ( ˆ ρ k Λ ) =
1, and with the same interpretationof having one particle per volume V as above. The vectorpotential that corresponds to the plane-wave one-photonstate can be calculated as A k Λ ( x ) ∶= ⟨ ∣ ˆ A ( x )∣ k Λ ⟩ = N k u k Λ = √ πckV ε k Λ e i k · x . (A.12)2
3. Twisted-wave photon states
Twisted photons are cylindrical symmetric modes andtherefore need to be normalized to a cylindrical volume V = πR L z with radius R and height L z along the z -axis.The twisted photon states that we defined in Eq. (28) areorthonormalized in the following sense: ⟨ κ ′⊥ κ ′∥ m ′ Λ ′ ∣ κ ⊥ κ ∥ m Λ ⟩= π RL z δ ( κ ′∥ − κ ∥ ) δ ( κ ′⊥ − κ ⊥ ) δ m ′ m δ Λ ′ Λ , (A.13)which implies that ⟨ κ ⊥ κ ∥ m Λ ∣ κ ⊥ κ ∥ m Λ ⟩ =
1. To prove thisnormalization we make use of the identity for the radialdelta function lim κ ′⊥ → κ ⊥ δ ( κ ′⊥ − κ ⊥ ) = Rπ , (A.14)where R is the (infinite) radius of the cylindrical nor-malization volume. The above identification was proven,e.g., in Refs. [29, 38]. Moreover, for the longitudinalmomentum delta function we employ the usual relation δ ( κ ∥ = ) = L z / π , where L z is the height of the quanti-zation cylinder.The probability density that can be attributedto the twisted one-particle states n κ ⊥ κ ∥ m Λ ( x ) = tr ( ˆ n ( x ) ˆ ρ κ ⊥ κ ∥ m Λ ) turns out the be not spatially constant,but instead is given by n κ ⊥ κ ∥ m Λ ( x ) = κ ⊥ L z R J m ( κ ⊥ x ⊥ ) , (A.15)where J m are the Bessel functions of the first kind [41].The spatially averaged probability density ⟨ n ⟩ ∶= V ∫ d x n ( x ) = V trˆ ρ (A.16)which enters the definition of the cross section, Eq. (6),is just proportional to the trace of the density opera-tor because the position eigenstates form a complete ba-sis. That means, also for twisted photons a normalizeddensity operator correspond to one particle per volume V = πL z R . The spatially averaged probability density can also be calculated directly from Eq. (A.15) as ⟨ n κ ⊥ κ ∥ m Λ ⟩ = V ∫ d ϕ d z d x ⊥ x ⊥ κ ⊥ L z R J m ( κ ⊥ x ⊥ )= V πR R ∫ d x ⊥ x ⊥ κ ⊥ J m ( κ ⊥ x ⊥ ) , (A.17)and by approximating the radial integral over x ⊥ for large R → ∞ by using the asymptotic expansion of the Besselfunction for large arguments, J m ( x ) ≈ √ / πx cos ( x − mπ / − π / ) [38, 41], yielding R ∫ d x ⊥ κ ⊥ x ⊥ J m ( κ ⊥ x ⊥ ) ≈ Rπ . (A.18)The vector potential that corresponds to the twistedone-photon states is given by A κ ⊥ κ ∥ m Λ ( x ) = ⟨ ∣ ˆ A ( x )∣ κ ⊥ κ ∥ m Λ ⟩= √ π c ωL z R ∫ d k ⊥ ( π ) a κ ⊥ m ( k ⊥ ) ε k Λ e i k ⊥ · x ⊥ + iκ ∥ z . (A.19)Except for the different normalization factor in front ofthe integral, this coincides with the vector potential em-ployed in Ref. [27] to define the twisted light. It was ar-gued in [27] that A µκ ⊥ κ ∥ m Λ ( x ) describes a beam with well-defined projection of total angular momentum m . Theproduct of the amplitude a κ ⊥ m ( k ⊥ ) and the plane-wavepolarization vector ε k Λ was shown to be an eigenfunc-tion of the z -component of the total angular momentumoperator ˆ J z with the eigenvalue m .Performing the momentum integrations in the aboveFourier integral we obtain for the twisted-wave vectorpotential A κ ⊥ κ ∥ m Λ ( x ) = √ π c ωL z R √ κ ⊥ π e iκ ∥ z × ∑ m = , ± (− ) m s c m s ( Λ ) J m − m s ( κ ⊥ x ⊥ ) e i ( m − m s ) ϕ η m s (A.20)where the sum over m s runs over all possible projec-tions of the photon spin angular momentum onto the z -direction and accounts for the coupling of (the projec-tions of) orbital angular momentum ( m (cid:96) ) and spin an-gular momentum ( m s ) to the total angular momentum m = m (cid:96) + m s . This representation of the twisted wave vec-tor potential employs the unit vectors η = ( , , ) T and η ± = ( , ± i, ) T /√
2, and the coefficients c = − √ sin θ k and c ± = ( ± Λ cos θ k ) with the momentum cone open-ing angle tan θ k = κ ⊥ / κ ∥ . [1] A. H. Compton, “A quantum theory of the scattering ofx-rays by light elements,” Phys. Rev. , 483 (1923). [2] O. Klein and Y. Nishina, “ ¨Uber die Streuung von Strahlung durch freie Elektronen nach der neuen rela-tivistischen Quantenmechanik nach Dirac,” Z. Phys. ,853 (1929).[3] J. D. Jackson, Klassische Elektrodynamik , 2nd ed. (Wal-ter de Gruyter, Berlin, New York, 1983).[4] W. Greiner,
Quantentheorie: Spezielle Kapitel , 3rd ed.,Theoretische Physik, Vol. 4A (Verlag Harri Deutsch,1989).[5] W. B. Berestetzki, E. M. Lifschitz, and L. P. Pitajewski,
Relativistische Quantentheorie , Lehrbuch der Theoreti-schen Physik, Vol. IV (Akademie Verlag, Berlin, 1980).[6] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw,and J. P. Woerdman, “Orbital angular momentum oflight and the transformation of Laguerre-Gaussian lasermodes,” Phys. Rev. A , 8185 (1992).[7] A. T. O’Neil, I. MacVicar, L. Allen, and M. Padgett,“Intrinsic and extrinsic nature of the orbital angular mo-mentum of a light beam,” Phys. Rev. Lett. , 053601(2002).[8] G. Molina-Terriza, J. P. Torres, and L. Torner, “Twistedphotons,” Nature Phys. , 305 (2007).[9] A. M. Yao and M. J. Padgett, “Orbital angular momen-tum: origins, behavior and applications,” Advances inOptics and Photonics , 161 (2011).[10] D. L. Andrews and M. Babiker, eds., The Angular Mo-mentum of Light (Cambridge University Press, 2013).[11] K. Blum,
Density Matrix Theory and Applications , 3rded., Springer Series on Atomic, Optical, and PlasmaPhysics, Vol. 64 (Springer-Verlag, Berlin, Heidelberg,New York, 2012).[12] T. J. Englert and E. A. Rinehart, “Second-harmonic pho-tons from the interaction of free electrons with intenselaser radiation,” Phys. Rev. A , 1539 (1983).[13] D. Cojoc, B. Kaulich, A. Carpentiero, S. Cabrini,L. Businaro, and E. Di Fabrizio, “X-ray vortices withhigh topological charge,” Microelectronic Engineering , 1360 (2006).[14] M. Z¨urch, C. Kern, P. Hansinger, A. Dreischuh, andCh. Spielmann, “Strong-field physics with singular lightbeams,” Nature Phys. , 743 (2012).[15] E. Hemsing, A. Knyazik, M. Dunning, D. Xiang, A.Marinelli, C. Hast, and J. B. Rosenzweig, “Coherentoptical vortices from relativistic electron beams,” NaturePhys. , 549 (2013).[16] J. Bahrdt, K. Holldack, P. Kuske, R. M¨uller, M. Scheer,and P. Schmid, “First observation of photons carrying or-bital angular momentum in undulator radiation,” Phys.Rev. Lett. , 034801 (2013).[17] G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond,E. Frumker, R. W. Boyd, and P. B. Corkum, “Creat-ing high-harmonic beams with controlled orbital angularmomentum,” Phys. Rev. Lett. , 153901 (2014).[18] J. Arlt and K. Dholakia, “Generation of high-order besselbeams by use of an axicon,” Opt. Commun. , 297(2000).[19] A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, “En-gineering the size of dark core of an optical vortex,” Opt.Las. in Engineering , 276 (2010).[20] V. V. Balashov, A. N. Grum-Grzhimailo, and N. M.Kabachnik, Polarization and Correlation Phenomena inAtomic Collisions (Springer, 2000).[21] A. Surzhykov, D. Seipt, V. G. Serbo, and S. Fritzsche,“Interaction of twisted light with many-electron atomsand ions,” Phys. Rev. A , 013403 (2015). [22] J. R. Taylor, Scattering Theory: The Quantum Theoryof Nonrelativistic Collisions (John Wiley & Sons, 1972).[23] L. D. Landau and E. M. Lifschitz,
Klassische Feldtheo-rie , 12th ed., Lehrbuch der Theoretischen Physik, Vol. 2(Akademie Verlag, 1992).[24] By using the relation for the energy delta functionlim E ′ i → E i πδ ( E i − E ′ i ) = lim E ′ i → E i T / ∫ − T / d t e i ( E i − E ′ i ) t = T in the limit T → ∞ [40].[25] E. Noether, “Invariante Variationsprobleme,” Nachr.Ges. Wiss. G¨ottingen, Math.-Phys. Kl. , 235–257(1918), english translation: M. A. Travel, Transport The-ory and Statistical Physics , 183 (1971).[26] W. H. McMaster, “Matrix representation of polariza-tion,” Rev. Mod. Phys. , 8 (1961).[27] O. Matula, A. G. Hayrapetyan, V. G. Serbo,A. Surzhykov, and S. Fritzsche, “Atomic ionization ofhydrogen-like ions by twisted photons: angular distribu-tion of emitted electrons,” J. Phys. B , 205002 (2013).[28] U. D. Jentschura and V. G. Serbo, “Generation of high-energy photons with large orbital angular momentum byCompton backscattering,” Phys. Rev. Lett. , 013001(2011).[29] I. P. Ivanov, “Colliding particles carrying nonzero orbitalangular momentum,” Phys. Rev. D , 093001 (2011).[30] D. Seipt, A. Surzhykov, and S. Fritzsche, “Structuredx-ray beams from twisted electrons by inverse Comptonscattering of laser light,” Phys. Rev. A , 012118 (2014).[31] H. M. Scholz-Marggraf, S. Fritzsche, V. G. Serbo,A. Afanasev, and A. Surzhykov, “Absorption of twistedlight by hydrogenlike atoms,” Phys. Rev. A , 013425(2014).[32] V. G. Serbo, I. V. Ivanov, S. Fritzsche, D. Seipt, and A.Surzhykov, “Scattering of twisted relativistic electrons byatoms”, submitted.[33] I. P. Ivanov, “Measuring the phase of the scattering am-plitude with vortex beams,” Phys. Rev. D , 076001(2012).[34] S. Lloyd, M. Babiker, and J. Yuan, “Quantized orbitalangular momentum transfer and magnetic dichroism inthe interaction of electron vortices with matter,” Phys.Rev. Lett. , 074802 (2012).[35] R. Van Boxem, B. Partoens, and J. Verbeeck, “Ruther-ford scattering of electron vortices,” Phys. Rev. A ,032715 (2014).[36] R. Van Boxem, B. Partoens, and J. Verbeeck, “Inelas-tic electron-vortex-beam scattering,” Phys. Rev. A ,032703 (2015).[37] G. F. Quinteiro, D. E. Reiter, and T. Kuhn, “Formula-tion of the twisted-light–matter interaction at the phasesingularity: The twisted-light gauge,” Phys. Rev. A ,033808 (2015).[38] U. D. Jentschura and V. G. Serbo, “Compton upconver-sion of twisted photons: backscattering of particles withnon-planar wave functions,” Eur. Phys. J. C , 1571(2011).[39] R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Gener-ating superpositions of higher–order bessel beams,” Opt.Express , 23389 (2009).[40] M. E. Peskin and D. V. Schroeder, An Introductionto Quantum Field Theory (Addison-Wesley PublishingCompany, 1995).[41] G. N. Watson,
A treatise on the theory of Bessel functions4